FE Mechanical flashcards

34 cards. Read the term and the definition here. The deck at the top of this page is optional.

Chain rule

Derivative of a composite function f(g(x)) equals f'(g(x)) times g'(x). Essential for differentiating nested expressions such as sin(x^2) or e^(3x).

L'Hôpital's rule

For limits producing 0/0 or ∞/∞, the limit of f/g equals the limit of f'/g' if that limit exists. Apply repeatedly if needed.

Eigenvalue equation

For square matrix A, eigenvalues λ satisfy det(A − λI) = 0. Eigenvectors then solve (A − λI)x = 0. Diagonal entries of a triangular matrix are its eigenvalues.

Dot product

a·b = |a||b|cosθ = a_x b_x + a_y b_y + a_z b_z. Result is a scalar; zero when vectors are perpendicular. Used for work and projections.

Cross product

a×b is a vector perpendicular to both a and b with magnitude |a||b|sinθ. Direction follows right-hand rule. Used for moments and areas.

Newton-Raphson method

Iterative root finding: x_{n+1} = x_n − f(x_n)/f'(x_n). Converges quadratically near a simple root but can diverge if the derivative is near zero.

Binomial distribution

Probability of k successes in n independent trials with success probability p: P(X=k) = C(n,k) p^k (1−p)^(n−k). Mean np, variance np(1−p).

Normal standardization

Convert X ~ N(μ, σ) to standard normal Z = (X − μ)/σ. Then use standard normal tables. Approximately 95% of values lie within μ ± 2σ.

Central limit theorem

For large sample size n, the sample mean is approximately normally distributed with mean μ and standard deviation σ/√n, regardless of the population distribution.

Confidence interval for mean

With known σ: x̄ ± z_(α/2) σ/√n. With unknown σ and small n: x̄ ± t_(α/2, n−1) s/√n. Width decreases as n increases.

Coefficient of determination

R^2 is the proportion of variance in the dependent variable explained by the regression. Ranges 0 to 1; higher values indicate better fit.

Combinations formula

Number of ways to choose r items from n without regard to order: nCr = n!/[r!(n−r)!]. For order-sensitive selections use permutations nPr = n!/(n−r)!.

Paramount public safety

Under NCEES Model Rules, engineers shall hold paramount the safety, health, and welfare of the public. This duty overrides obligations to employers or clients.

Responsible charge

A PE may seal only documents prepared by them or under their direct supervisory control. Sealing work outside responsible charge is a violation of professional conduct rules.

Conflict of interest disclosure

Engineers must disclose known or potential conflicts to clients or employers. Accepting compensation from multiple parties for the same service requires full disclosure and consent.

Qualifications-based selection

Professional engineering services are typically procured based on qualifications and competence rather than low bid. Price negotiation follows selection of the most qualified firm.

Negligence elements

A negligence claim requires duty, breach of duty, causation (actual and proximate), and damages. All four must be shown for liability.

Effective annual interest rate

i_eff = (1 + i_nom/m)^m − 1 where m is compounding periods per year. Converts nominal rate to true annual yield; increases with m.

Capital recovery factor

(A/P, i, n) = i(1+i)^n / [(1+i)^n − 1]. Converts a present amount to an equivalent uniform annual series over n periods at rate i.

Internal rate of return

The interest rate i that makes net present worth equal zero. Accept an independent project if IRR ≥ MARR. Multiple sign changes can yield multiple IRRs.

Straight-line depreciation

Annual depreciation D = (B − S)/n where B is first cost, S salvage value, n useful life. Book value after k years = B − kD.

Breakeven quantity

Q = FC/(p − v) where FC is fixed cost, p unit price, v variable cost per unit. Produces zero profit; above Q the operation is profitable.

Resultant of distributed load

Equivalent force equals the area under the load diagram and acts through the centroid of that area. For uniform load w over length L: R = wL at L/2.

Zero-force truss member

A member is zero-force if at a joint with no external load it is the only member in one direction, or if two noncollinear members meet with no load and no support reaction.

Parallel axis theorem

I = I_c + A d^2 where I_c is moment of inertia about centroidal axis, A area, d distance between parallel axes. Used for composite areas.

Beam shear-moment relationship

dV/dx = −w(x) and dM/dx = V. Bending moment is maximum where shear is zero or changes sign. Area under shear diagram equals change in moment.

Coefficient of restitution

e = (v2' − v1')/(v1 − v2) for direct central impact. e = 1 perfectly elastic, e = 0 perfectly plastic (stick together). Momentum is conserved in both.

Projectile range

For level ground, range R = v0^2 sin(2θ)/g. Maximum range occurs at θ = 45°. Time of flight t = 2v0 sinθ/g.

Normal acceleration

a_n = v^2/ρ = ω^2 r directed toward the center of curvature. Tangential acceleration a_t = dv/dt. Total acceleration magnitude = √(a_t^2 + a_n^2).

Natural frequency

Undamped spring-mass system: ω_n = √(k/m) rad/s, f_n = ω_n/(2π) Hz. Damped frequency ω_d = ω_n√(1−ζ^2). Critical damping c_c = 2√(km).

Torsion formula

Shear stress in a circular shaft: τ = Tc/J where c is outer radius, J polar moment of inertia. Angle of twist φ = TL/(JG). J = πc^4/2 for solid shaft.

Flexure formula

Bending stress σ = My/I where y is distance from neutral axis, I area moment of inertia. Maximum stress at outer fiber: σ_max = Mc/I = M/S.

Principal stresses

For plane stress: σ_1,2 = (σ_x+σ_y)/2 ± √[((σ_x−σ_y)/2)^2 + τ_xy^2]. Maximum in-plane shear = √[((σ_x−σ_y)/2)^2 + τ_xy^2].

Euler buckling load

P_cr = π^2 EI/(KL)^2 where K depends on end conditions: pinned-pinned 1.0, fixed-fixed 0.5, fixed-pinned 0.7, fixed-free 2.0. Valid for slender columns.